COUPLED FIXED POINTS AND COUPLED BEST PROXIMITY POINTS IN MODULAR FUNCTION SPACES

A. Ilchev, B. Zlatanov

IJPAM: Volume 118, No. 4 (2018)

Title

COUPLED FIXED POINTS AND COUPLED BEST
PROXIMITY POINTS IN MODULAR FUNCTION SPACES

Authors

Atanas Ilchev

, Boyan Zlatanov

$^{1,2}$ University of Plovdiv ``Paisii Hilendarski''
Plovdiv, BULGARIA

Abstract

We generalize the notion of coupled fixed points and coupled best proximity points in the context of modular function spaces. We have found sufficient conditions for the existence and uniqueness of coupled fixed points and coupled best proximity points in modular function spaces.

We present an application of the main result, that generalizes well known examples about coupled best proximity points.

History

Received: 2018-04-17
Revised: 2018-05-8
Published: May 9, 2018

AMS Classification, Key Words

AMS Subject Classification: 47H10, 54H25, 45D05, 46A80
Key Words and Phrases: coupled fixed points, coupled best proximity points, modular function space

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Bibliography

1: T.G. Bhaskar, V. Lakshmikantham, Fixed Point Theorems in Partially Ordered Metric Spaces and Applications, Nonlinear Anal., 65(7) (2006), 1379-1393.http://dx.doi.org/10.1016/j.na.2005.10.017.
2: A. Eldred, P. Veeramani, Existence and Convergence of Best Proximity Points, J. Math. Anal. Appl., 323(2) (2006), 1001-1006.http://dx.doi.org/10.1016/j.jmaa.2005.10.081.
3: D. Guo, V. Lakshmikantham, Coupled Fixed Points of Nonlinear Operators with Applications, Nonlinear Anal., 11(5) (1987), 623-632. http://dx.doi.org/10.1016/0362-546X(87)90077-0.
4: Animesh Gupta, S.S. Rajput, P.S. Kaurav, Coupled Best Proximity Point Theorem in Metric Spaces, Int. J. Anal. Appl.,4(2) (2014), 201-215.
5: J. Harjani, B. López, K. Sadarangani, Fixed Point Theorems for Mixed Monotone Operators and Applications to Integral Equations, Nonlinear Anal., 74(5) (2011), 1749–-1760. http://dx.doi.org/10.1016/j.na.2010.10.047.
6: A. Ilchev, B. Zlatanov, Error Estimates for Approximation of Coupled Best Proximity Points for Cyclic Contractive Maps, Applied Mathematics and Computation, 290 (2016), 412–-425. http://dx.doi.org/10.1016/j.amc.2016.06.022.
7: A. Ilchev, B. Zlatanov, Fixed and Best Proximity Points for Kannan Cyclic Contractions in Modular Function Spaces, Journal of Fixed Point Theory Applications, 19(4) (2017), 2873–-2893. http://dx.doi.org/10.1007/s11784-017-0459-4.
8: H. Işik, D. Türkoğlu, Coupled Fixed Point Theorems for New Contractive Mixed Monotone Mappings and Applications to Integral Equations, Filomat, 28(6) (2014), 1253-–1264.
9: M.A. Khamsi, W.M. Kozlowski, Fixed Point Theory in Modular Function Spaces, Birhäuser, Heidelberg New York Dordrecht London (2015).
10: M.A. Khamsi, W. Kozlowski, S. Reich, Fixed Point Theory in Modular Function Spaces, Nonlinear Anal., 14(11) (1990), 935-953.http://dx.doi.org/10.1016/0362-546X(90)90111-S.
11: M.A. Khamsi, W.M. Kozlowski, Chen Shutao, Some Geometrical Properties and Fixed Point Theorems in Orlicz Spaces, J. Math. Anal. Appl., 155(2) (1991), 393-412. http://dx.doi.org/10.1016/0022-247X(91)90009-O.
12: W.A. Kirk, P. Srinivasan, P. Veeramani, Fixed Points for Mappings Satisfying Cyclical Contractive Conditions, Fixed Point Theory, 4(1) (2003), 79-89.
13: W.M. Kozlowski, Notes on modular function spaces I, Comment. Math., 28(1) (1988), 87-100.
14: W.M. Kozlowski, Notes on modular function spaces II, Comment. Math., 28(1) (1988), 101-116.
15: W.M. Kozlowski, Modular Function Spaces, Marcel Dekker, Inc., New York and Basel (1988).
16: W.M. Kozlowski, Advancements in Fixed Point Theory in Modular Function Spaces, Arabian Journal of Mathematics, 1(4) (2012), 477-494.http://dx.doi.org/10.1007/s40065-012-0051-0.
17: J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces I. Sequence Spaces, Springer–-Verlag, Berlin Heidelberg New York, (1977).
18: J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces II. Sequence Spaces, Springer–-Verlag, Berlin Heidelberg New York, (1979).
19: J. Musielak, W. Orlicz, On Modular Spaces, Stud. Math., 18(1) (1959), 49-65. http://dx.doi.org/10.4064/sm-18-1-49-65.
20: H. Nakano, Modular Semi-Ordered Spaces, Maruzen, Tokyo (1950).
21: M. Rao, Z. Ren, Theory of Orlicz Spaces, Marcel Dekker, Inc., New York Basel Hong Kong, (1991).
22: M. Rao, Z. Ren, Applications of Orlicz Spaces, Marcel Dekker, Inc., New York Basel, (2002).
23: W. Sintunavarat, P. Kumam, Coupled Best Proximity Point Theorem in Metric Spaces, Fixed Point Theory Appl., 2012/1/93 (2012).http://dx.doi.org/10.1186/1687-1812-2012-93
24: B. Zlatanov, Best Proximity Points in Modular Function Spaces, Arabian Journal of Mathematics, 4(3) (2015), 215-227.http://dx.doi.org/10.1007/s40065-015-0134-9.
25: B. Zlatanov, Error Estimates for Approximating of Best Proximity Points for Cyclic Contractive Maps, Carpathian J. Math., 32(2) (2016), 265-270.

How to Cite?

DOI: 10.12732/ijpam.v118i4.10 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2018
Volume: 118
Issue: 4
Pages: 957 - 977

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